Asymptotics of Chebyshev Polynomials, I. Subsets of R

We consider Chebyshev polynomials, Tn(z), for infinite, compact sets e ⊂ R (that is, the monic polynomials minimizing the sup–norm, ‖Tn‖e, on e). We resolve a 45+ year old conjecture of Widom that for finite gap subsets of R, his conjectured asymptotics (which we call Szegő–Widom asymptotics) holds. We also prove the first upper bounds of the form ‖Tn‖e ≤ QC(e) (where C(e) is the logarithmic capacity of e) for a class of e’s with an infinite number of components, explicitly for those e ⊂ R that obey a Parreau–Widom condition.